On Janowski Type Harmonic Meromorphic Functions with respect to Symmetric Point

One of the contemporary developments in Mathematics is the solicitations of harmonic analysis in other fields. Like various other fields, it has immensely influenced and nurtured the branch of geometric function theory. Jahangiri et al. [1] defined and studied a subclass of harmonic and univalent functions. Another example of such work would be an article of Porwal and Dixit [2], who used a certain convolution operator involving hypergeometric functions to define a class of univalent functions. As a consequence, many mathematicians generalized many ideas of this field and various important results with the help of some operators; the work of Porwal et al. [3], Porwal et al. [4], and Porwal and Dixit [5] are worth mentioning here. Recently, some subclasses of harmonic functions were investigated by Arif et al. [6] and Khan et al. [7]. To start with, we give preliminaries which will be useful in understanding the concepts of this research. A real-valued function uðx, yÞ is said to be harmonic in a domain D ⊂C if it has a continuous second partial derivative and satisfy the Laplace’s equation


Introduction and Definitions
One of the contemporary developments in Mathematics is the solicitations of harmonic analysis in other fields. Like various other fields, it has immensely influenced and nurtured the branch of geometric function theory. Jahangiri et al. [1] defined and studied a subclass of harmonic and univalent functions. Another example of such work would be an article of Porwal and Dixit [2], who used a certain convolution operator involving hypergeometric functions to define a class of univalent functions. As a consequence, many mathematicians generalized many ideas of this field and various important results with the help of some operators; the work of Porwal et al. [3], Porwal et al. [4], and Porwal and Dixit [5] are worth mentioning here. Recently, some subclasses of harmonic functions were investigated by Arif et al. [6] and Khan et al. [7]. To start with, we give preliminaries which will be useful in understanding the concepts of this research.
A real-valued function uðx, yÞ is said to be harmonic in a domain D ⊂ ℂ if it has a continuous second partial derivative and satisfy the Laplace's equation A continuous complex-valued function f = u + iv is said to be harmonic in a complex domain U if both its real and imaginary parts are real harmonic in U. In any simply connected domain U ⊂ ℂ, one can write f = h + g, where h and g are analytic in U. The class of such functions is denoted by H . The condition jh ′ ðzÞj > jg ′ ðzÞj is necessary and sufficient for f to be locally univalent and sense preserving in U, see [8]. There are different papers on univalent harmonic functions defined in unit disc D = fz : jzj < 1g, for details, see [9][10][11][12][13][14]. For z ∈ D * = D \ f0g, in the punctured open unit disc and let M H denote the class of functions which are harmonic in D * where h is analytic in D * and has a simple pole at the origin with residue 1, while g is analytic in D. The class M H was studied in [15][16][17]. Furthermore, denoted by M H , a subclass of M H , consisting a functions of the form which are harmonic univalent in punctured unit disc D * : For functions f ∈ M H given by (2) and F ∈ M H given by we recall the Hadamard product (or convolution) of f and F by In terms of the Hadamard product (or convolution), we choose F as a fixed function in H such that ðf * FÞð zÞ exists for any f ∈ H , and for various choices of F, we get different linear operators which have been studied in the recent past.
Recently, Khan et al. [18] introduced and studied a class of meromorphic starlike functions with respect to symmetric point in circular domain i.e., Motivated from the above discussion on harmonic functions and class of meromorphic starlike functions with respect to symmetric point, we introduced the class of meromorphic harmonic univalent functions as: where the symbol } ≺ } represent well-known subordination and or equivalently

Main Results
Theorem 1. Let f = h + g be of the form (2) and satisfies the condition with then f is harmonic univalent sense-preserving in D * and f ∈ M * * H ½A, B: where we have used (10) and this shows that the function is univalent.
Now to show f ðzÞ is sense-preserving harmonic mapping in D * , consider This shows that f is sense-preserving. Now, to show that f ∈ M * * H ½A, B from (9), it is enough to show that

Journal of Function Spaces
For this, consider α n a n j j+ β n b n j j ½ − 1 ( ) Hence, complete the proof.

Example 2. The meromorphic univalent function
such that ∑ ∞ n=1 ðjx n j + jy n jÞ = 1, we have α n a n j j + β n b n j j ð Þ = 〠 ∞ n=1 x n j j + y n j j ð Þ= 1: Thus, f ∈ M * * H ½A, B and above coefficient bound given in (10) is sharp for this function. α n a n j j + β n b n j j ≤ 1, ð18Þ with Proof. The proof is similar to Theorem 1, so omitted.